Linear Algebra

Section II. Linear Geometry of n-Space 39II.2 Length and Angle MeasuresWe’ve translated the first section’s results about solution sets into geometricterms for insight into how those sets look. But we must watch out not to bemislead by our own terms; labeling subsets of R k of the forms {⃗p + t⃗v ∣ ∣ t ∈ R}and {⃗p + t⃗v + s ⃗w ∣ ∣ t, s ∈ R} as “lines” and “planes” doesn’t make them act likethe lines and planes of our prior experience. Rather, we must ensure that thenames suit the sets. While we can’t prove that the sets satisfy our intuition —we can’t prove anything about intuition — in this subsection we’ll observe thata result familiar from R 2 and R 3 , when generalized to arbitrary R k , supportsthe idea that a line is straight and a plane is flat. Specifically, we’ll see how todo Euclidean geometry in a “plane” by giving a definition of the angle betweentwo R n vectors in the plane that they generate.2.1 Definition The length of a vector ⃗v ∈ R n is this.√‖⃗v ‖ = v1 2 + · · · + v2 n2.2 Remark This is a natural generalization of the Pythagorean Theorem. Aclassic discussion is in [Polya].We can use that definition to derive a formula for the angle between twovectors. For a model of what to do, consider two vectors in R 3 .⃗v⃗uPut them in canonical position and, in the plane that they determine, considerthe triangle formed by ⃗u, ⃗v, and ⃗u − ⃗v.Apply the Law of Cosines, ‖⃗u − ⃗v ‖ 2 = ‖⃗u ‖ 2 + ‖⃗v ‖ 2 − 2 ‖⃗u ‖ ‖⃗v ‖ cos θ, where θis the angle between the vectors. Expand both sides(u 1 − v 1 ) 2 + (u 2 − v 2 ) 2 + (u 3 − v 3 ) 2= (u 2 1 + u 2 2 + u 2 3) + (v 2 1 + v 2 2 + v 2 3) − 2 ‖⃗u ‖ ‖⃗v ‖ cos θand simplify.θ = arccos( u 1v 1 + u 2 v 2 + u 3 v 3‖⃗u ‖ ‖⃗v ‖)